Question

Find the limit:
`\lim_(a x \to 0) ((x + ax)^(2)-2(x + ax) + 1 - (x^(2) - 2x + 1))/(ax)`

Answer 1

I'll let h = ax, so the limit is

`\displaystyle\lim_(h\to0)\frac{(x+h)^2-2(x+h)+1-(x^2-2x+1)}h`

i.e. the derivative of
`x^2-2x+1`.

Expand the numerator to see several terms that get eliminated:

`(x+h)^2-2(x+h)+1-(x^2-2x+1)=x^2+2xh+h^2-2x-2h+1-x^2+2x-1=2xh+h^2-2h`

So we have

`\displaystyle\lim_(h\to0)\frac{2xh+h^2-2h}h`

Since h ≠ 0 (because it is approaching 0 but never actually reaching 0), we can cancel the factor of h in both numerator and denominator, then plug in h = 0:

`\displaystyle\lim_(h\to0)(2x+h-2)=\boxed{2x-2}`

Answer 2

2x-2

Explanation:

lim ax goes to 0 ( x+ ax)^2 -2 ( x+ax) +1 - ( x^2 -2x+1)

--------------------------------------------------

ax

Simplify the numerator by foiling the first term and distributing the minus signs

x^2+ 2ax^2 + a^2 x^2 -2x-2ax +1 - x^2 +2x-1

--------------------------------------------------

ax

Combine like terms

2ax^2 + a^2 x^2 -2ax

--------------------------------------------------

ax

Factor out ax

ax( 2x + ax -2)

----------------------

ax

Cancel ax

2x + ax -2

Now take the limit

lim ax goes to 0 ( 2x + ax -2)

2x +0-2

2x -2